The present invention relates to a position control system having force as a manipulated variable and position as a controlled variable.
In the position control system as shown in FIG. 1, a movable table 12, whose position is to be controlled precisely, is driven by a motor 11 through ball screw 15. The motor 11 is driven by a power amplifier 19, and a force "f" proportional to the force command "u", which is an input of the power amplifier 19, is applied on the movable table 12. On the movable table 12, a corner cube 14 and an accelerometer 13 are fixed. The position "x" of the movable table 12 is measured by the laser measuring machine 16 and corner cube 14, and the velocity "v" is obtained from the position "x", using velocity arithmetic circuit 17. Further, the acceleration "a" is measured by the accelerometer 13. The controller 18 uses the position command "r" and the position "x", the velocity "v" and the acceleration "a" as inputs and issues the force command "u" so that the position "x" follows the position command "r". In such a position control system, the present invention proposes a position control method to define the composition of the controller 18.
Suppose that, in the above position control system, the response of the motor 11 and the power amplifier 19 are sufficiently quick and linear. Then, the relation between the force "f" applied on the movable table 12 and the force command "u" output by the controller 18 is given by the following equation: EQU f=K.sub.f u (1),
where K.sub.f is a proportional constant.
If the response of the position control system is slow and the mechanism (including the motor 11, ball screw 15 and movable table 12) is a rigid body, the relation between the position "x" of the movable table 12 and the applied force "f" is obtained by: EQU Mx=f (2),
where M is the mass of the movable portion. This relation is shown as a block diagram in FIG. 2. Because this system is a quadratic system, the position "x" and the velocity "v" will suffice to express the quantity of state. It is known that the position control system with adequate characteristics can be composed by negative feedback of the position "x" and the velocity "v" as it is often practiced in the past. The control system is expressed by the following equation: EQU u=Kv {Kp (r-x)-v} (3),
where Kp and Kv are the feedback gains. This relation is shown in FIG. 3 as a block diagram.
In general, it is expected that the position control system has higher accuracy and higher response with the increase of the gains Kp and Kv, whereas there is the limitation to this because the behavior of the mechanism cannot be assumed as that of a rigid body when the response of the position control system becomes quicker, and vibration characteristics must be taken into consideration.
When the vibration characteristics of the mechanism is considered, the relation between the position and the force is shown by the block diagram of FIG. 4. Here, .omega. and .zeta. are angular natural frequency and damping ratio respectively. In general, the damping ratio of the mechanism is small, and the vibration characteristics become unstable with the increase of gain.
Thus, there arises a problem in the conventional control method when vibration characteristics are no longer negligible. This will be explained below by an example.
FIG. 5 is a block diagram showing a velocity control system by the conventional control method. FIG. 6 shows the root loci of the velocity control system as shown FIG. 5, and v.sub.r is the velocity command.
The property of the feedback control system is determined by the pole of the closed loop. FIG. 6 shows the locus (root locus) of the pole of the closed loop when the gain K.sub.v is gradually increased from 0 in this system. Here, it is assumed that .omega.=1 and .zeta..apprxeq.0. As it is evident from FIG. 6, the vibration characteristics moves rightward, i.e. toward unstability with the increase of gain. Because the vibration roots on the imaginary axis are non-damping, i.e. instability limit, the gain must be smaller than the case of this condition. In general, the damping ratio of the mechanism is small, and it reaches the stability limit. This makes it impossible to increase the gain in the control system and to provide sufficient accuracy and response. This result does not vary significantly when negative feed back of the acceleration "a" is added in the system of FIG. 5.
In the position control system, response is slower than in the velocity control system, and sufficient characteristics can not be obtained by the conventional control method of equation (3).